Complements #7 The Pythagorean Theorem

The Pythagorean Theorem is possibly the first (and in my opinion, only) piece of beautiful maths in GCSE. Not only is it a majestic piece of mathematical wonder, it carries with it great history, real-life application, visual beauty and it opens the door to a wide range of geometry.

How tragic then, that it ends up being so poorly received by students, who often sit puzzling over where the hippopotamus is, which side is the ‘b side’, and trying to find the barely used square-root button on their calculator.

Pythagoras-2

Pythagoras is fairly lucky to have his name attached to the theorem, as the maths behind it had been used for many, many years before him. However kind of like patenting, he was the first to *formally* prove it, despite it being in use all over the place, and being independently discovered by various mathematicians all over the world.

pythagorean_squares_example

Ironically, the proof almost unanimously used in classrooms across the land, the so-called ‘Bride’s Chair’ proof (see above) is in fact NOT a Pythagorean Proof at all. It’s Euclid’s proof (almost). The *only* Pythagorean proof is this one:

Pythagoras-proof-anim.svg

This is the exact proof that Pythagoras came up with.

However there are literally hundreds of alternative proofs of the theorem to choose from should you so desire. And you should, because some are wonderfully clever.

Students will understand the properties of similar triangles, so why not prove it using those:

proof

Or a proof by tessellation

In fact, if you think carefully about the construction of a2 + b2 = c2, you may come to realise that you don’t even need squares at all. You just need similar shapes, whose side length is the length of the side of the right angled triangle (in fact, you don’t even need that last bit!)

circleindex

Any shape at all really… as long as each iteration is similar.

pythagoras-scaling-5

OK so now for some other interesting things linked (sort of) to the Pythagorean Theorem…

1. The Vecten Configuration

If you join the remaining vertices of the squares (brides chair), the new triangles are all of equal area to the area of the original triangle. This is in fact true for any triangle whatsoever (which I think is cooler than the Pythagorean Theorem… this one isn’t held back by specific triangles)

350-i-samearea

So in the picture above, all the turquoise triangles have equal area. Sadly we don’t use this fact in GCSE, but we should.

Following on from this, we can deduce the Finsler-Hadwiger Theorem, which looks like this:

FinslerHadwingerTheorem_1000

Put two squares together (ABCD and AB’C’D’) as shown, with one vertex touching, and you can create two triangles of equal area (also shown, DAB’ and BAD’), and by joining the midpoints of DB” and BD’ with the centres of the original squares, you get another square. Maybe it’s clearer on Geogebra…

geogeb

You can see the equal areas for both triangles in the side menu.

Finally, if you extend the sides of the squares from the Vecten configuration, you get Grebe’s Theorem (it’s not even on Wolfram!) which states that if you draw all those lines, you get two similar triangles that are homothetic (think “centre of enlargement”)

grebe

I could go on… but I won’t. Instead I’ll just show off my cool Pythagoras poster I made for my office:

ByJ4G_DIAAA2ESC

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One thought on “Complements #7 The Pythagorean Theorem

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